Indistinguishable particles in separate boxes

[Swamp Thing]

In this video at 02:41 we have two particles in two separate boxes. The voice-over says that the probability that particle P1 is at x1 and P2 is at x2 is equal to the probability that P1 is at x2 and P2 is at x1.



The ranges of x1 and x2 are non-overlapping and correspond to the two isolated boxes.

Now if P1 was emitted from a faraway galaxy on the left, and P2 was captured from a galaxy on the right, and they were then put into those boxes with no opportunity to swap places, would the above still apply? I remember reading somewhere (unfortunately can't recall the source) that even if there is no history of interaction, the wavefunction still has to have that kind of symmetry. If so, how can we describe how P1 can ever turn up in box 2 and vice versa? Is it something to do with tunneling from one box to another? Or is it just a thing to be assumed that has no valid intuitive picture?

 

[anuttarasammyak]

On the viewpoint of quantum field theory, wave-like probability amplitude presides the physics. Say we see ocean waves we can name this wave form Alice and that wave form Bob, but nature pay no attention to these characters we put on them.

 

[Swamp Thing]

But if one wave comes in from the east, and we trap (some of) its energy in a swimming pool... and we trap another wave from the west in another swimming pool, can we say that the wave from the east has some probability of turning up in the west pool -- even in some abstract sense?

Edit: assuming that there is a huge sandbar or peninsula (or even continent) that separates the two pools and also separates the water on the east from that on the west.

 

[PeroK]

Now if P1 was emitted from a faraway galaxy on the left, and P2 was captured from a galaxy on the right, and they were then put into those boxes with no opportunity to swap places, would the above still apply? I remember reading somewhere (unfortunately can't recall the source) that even if there is no history of interaction, the wavefunction still has to have that kind of symmetry. If so, how can we describe how P1 can ever turn up in box 2 and vice versa? Is it something to do with tunneling from one box to another? Or is it just a thing to be assumed that has no valid intuitive picture?

You've used a bit of sleight of hand there. You start by assuming that you have two isolated systems and then, just by uttering the magic words, you suddenly have one system of two identical particles.

You need to consider the system consistently throughout the experiment: either it's a system of two identical particles or not.

If two particles are far enough apart, then the system approximates to two isolated systems. Without that, you could never do quantum mechanics without considering all the particles in the universe.

 

[Swamp Thing]

You've used a bit of sleight of hand there.

Either the sleight of hand is on the part of the video creator, or I have misunderstood something they said. I'm watching it again, trying to be more careful.

When they first introduce the two boxes, they don't say exactly how the two particles got into those boxes in the first place. So I'm considering one possible history, i.e. that they started out isolated and ended up in those two boxes.

Another possible history is that they started out from the same source and were then separated and trapped in different boxes.

Would the wavefunction be different for these two cases? Specifically, would the symmetry requirement apply only to the second case?

 

[hilbert2]

If you consider the wave function of the whole universe, it's already antisymmetric in exchange of any two electrons. It doesn't matter if there's a 1000 light year distance or more between them.

 

[PeroK]

Either the sleight of hand is on the part of the video creator, or I have misunderstood something they said. I'm watching it again, trying to be more careful.

Those videos are brilliant, but they are not a complete, watertight description of experiments. How could they be? The answer is unlikely to be found in that video. What you need is more careful study of QM generally.

 

[PeterDonis]

how can we describe how P1 can ever turn up in box 2 and vice versa? Is it something to do with tunneling from one box to another?

No. Thinking of the Pauli exclusion principle in terms of "exchanging particles" can be misleading.

Try it this way: we have these two boxes, each one of which has one particle in it. Now we open the boxes and let the particles fly out, and we have two particle detectors, A and B. So we have three possible results for this experiment (assuming the particles can only go to one of the two detectors):

(1) Detector A detects two particles;

(2) Detector A and detector B each detect a particle;

(3) Detector B detects two particles.

The fact that the particles are indistinguishable means that we have no way of telling which particle came from which box. That means that, when we compute probability amplitudes for result #2 above, we have to add amplitudes for two possibilities:

(i) The particle from box 1 goes to detector A and the particle from box 2 goes to detector B;

(ii) The particle from box 1 goes to detector B and the particle from box 2 goes to detector A.

The other two results, #1 and #3, only have one possibility: both particles go to the same detector (A or B). Since we can't distinguish the particles, we can't distinguish two possibilities for this.

In other words, for all of the results, when computing amplitudes, we can only identify "the particle from box 1" and "the particle from box 2". We cannot say that "particle P1 came from box 1 and particle P2 came from box 2" is different from "particle P2 came from box 1 and particle P1 came from box 2". There are no such "labels" P1 and P2 on the particles. That is what "indistinguishable particles" means.

Now for the difference between bosons and fermions. For bosons, all three results are possible, and for result #2, the amplitudes for the two possibilities (i) and (ii) add. Also, the amplitudes for results #1 and #3 are twice as large as a calculation that ignored spin and statistics would suggest; that is because bosons have a larger amplitude to be in the same state.

For fermions, results #1 and #3 are not possible at all; this is the Pauli exclusion principle (two fermions can't be in the same state). In terms of amplitudes, the amplitude for the particle from box 1 to go to the detector is exactly canceled by the amplitude for the particle from box 2 to go to the same detector; they are equal in magnitude but opposite in sign. And for result #2 for fermions, the amplitudes for the two possibilities (i) and (ii) subtract. This is where "exchange" comes in: if we swap the two particles, which means swapping the two terms in the amplitude, the wave function has to flip sign. (The zero amplitude for results #1 and #3 can also be thought of this way: "swapping" the two particles here still leaves the same state since both particles are going to the same detector, and the only state vector that is equal to minus itself is the zero vector.)

 

[atyy]

Now if P1 was emitted from a faraway galaxy on the left, and P2 was captured from a galaxy on the right, and they were then put into those boxes with no opportunity to swap places, would the above still apply? I remember reading somewhere (unfortunately can't recall the source) that even if there is no history of interaction, the wavefunction still has to have that kind of symmetry. If so, how can we describe how P1 can ever turn up in box 2 and vice versa? Is it something to do with tunneling from one box to another? Or is it just a thing to be assumed that has no valid intuitive picture?

In principle one must include all other identical electrons in the universe in the quantum state of the electrons one is interested in. In practice, we can obtain excellent approximations by including in the quantum state only the electrons on Earth that we are interested in, and ignoring (for example) the electrons on the moon.

There is a detailed discussion in Shankar's Principles of Quantum Mechanics about this in the section "When can we ignore symmetrization and antisymmetrization" (p273-277 in the 1994 edition).

 

[vanhees71]

A very thorough discussion of the fact that you need not antisymmetrize wave functions where there are $N$ identical particles at positions $\vec{x}_1,\ldots,\vec{x}_N$ and $M$ other identical particles are at positions $x_{N+1},\ldots,x_{N+M}$ where the symmetrized or antisymmetrized wave functions $\psi(\vec{x}_1,\ldots \vec{x}_{N})$ have the property to vanish if at least one of these $\vec{x}_j's \notin D$ (where $D$ is some part of $\mathbb{R}^3$) while $\phi(\vec{x}_{N+1},\ldots,\vec{x}_{N+M}$ is an (anti)symmetrized wave function which vanishes if at least one of these $\vec{x}_k$ are $\notin D$, you can use the un(anti)symmetrized product
$$\Psi(\vec{x}_1,\ldots,\vec{x}_N,\vec{x}_{N+1},\ldots,\vec{x}_{N+M}) = \psi(\vec{x}_1,\ldots,\vec{x}_N) \phi(\vec{x}_{N+1},\ldots,\vec{x}_{N+M}).$$
The true wave function should of course be the (alternating) sum over all perturbations of positions $\vec{x}_1,\ldots,\vec{x}_{N+M}$, but in such a situation it's quite evident that all contributions in this (alternating) some not being equivalent with the above simple product are vanishing.